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MPI-I-1999-2-008

Cutting planes and the elementary closure in fixed dimension

Bockmayr, Alexander and Eisenbrand, Friedrich

MPI-I-1999-2-008. December 1999, 12 pages. | Status: available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:
The elementary closure $P'$ of a polyhedron $P$ is the intersection
of $P$ with all its Gomory-Chvátal cutting planes.
$P'$ is a rational polyhedron provided that $P$ is rational. The
known bounds for the number of inequalities defining $P'$ are
exponential, even in fixed dimension.
We show that the number of inequalities needed to describe the
elementary closure of a rational polyhedron is polynomially bounded
in fixed dimension.
If $P$ is a simplicial cone, we construct
a polytope $Q$, whose integral elements correspond to cutting planes
of $P$. The vertices of
the integer hull $Q_I$ include the facets of $P'$.
A polynomial upper bound on their number can be obtained by
applying a result of Cook et al.
Finally, we present a polynomial algorithm in varying dimension,
which computes cutting planes for a simplicial cone that
correspond to vertices of $Q_I$.
Acknowledgement:
References to related material:

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Hide details for BibTeXBibTeX
@TECHREPORT{BockmayrEisenbrandMPI-I-1999-2-008,
  AUTHOR = {Bockmayr, Alexander and Eisenbrand, Friedrich},
  TITLE = {Cutting planes and the elementary closure in fixed dimension},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-1999-2-008},
  MONTH = {December},
  YEAR = {1999},
  ISSN = {0946-011X},
}